12 Damage Location by Maximum Entropy Method on a Civil Structure 107 with AD 2 6 6 6 6 6 6 4 X1 1 X2 1 XN n X1 2 X2 2 XN n : : : X1 n 1 : : : X2 n 1 : : : : : : XN n 1 3 7 7 7 7 7 7 5 .nC1/ N ; b D 2 6 6 6 6 6 4 X1 X2 : : : Xn 1 3 7 7 7 7 7 5 .nC1/ 1 ; wD 2 6 6 6 4 !1 !2 : : : !N 3 7 7 7 5 N 1 After wis obtained from (12.2), Yˆ is estimated as b Y D N X jD1 !j .X/ Yj .X/ ; (12.3) where Y1(X), Y2(X), : : : , YN(X) are the corresponding observations to the N selected neighbors. Typically, the system of equations (12.2) is undetermined, and its solution is tackled via a constrained optimization technique of the family of least-squares. However, these methods produce some negative weights, which lacks physical meaning. An alternative that generates positive weights is obtained via the maximum-entropy principle [10], which can be written as: max p2RN C 2 4 H.!/ D N X jD1 !j .X/ln !j .X/ mj .X/ 3 5 (12.4a) Subject to the restrictions: N X jD1 !j .X/ QX j D0; N X jD1 !j .X/ D1; (12.4b) where mj(X) is a prior distribution that acts as a ‘first estimation’ for !j(X) and QX j DXj Xhas been introduced as a shifted measure for stability purposes. A typical prior distribution is the smooth Gaussian [12], defined as mj .X/ Dexp ˇjˇ ˇ ˇ ˇ ˇ ˇ Q X jˇ ˇ ˇ ˇ ˇ ˇ 2 ; (12.5) where: ˇj D =h2 j ; ” is a parameter that controls the radius of the Gaussian prior at Xj, and therefore its associated weight function; and hj is a n-dimensional Euclidean distance between neighbors that can be distinct for each Xj. In view of the optimization problem posed in (12.4a) for supervised learning, maximizing the entropy chooses the weight solution that commits the least to anyone in the databases samples [13]. The solution of the maximum entropy optimization problem is handled by using the procedure of Lagrange multipliers, which yields !j .X/ D Zj XI Z XI ; Zj XI Dmj .X/ exp e X i ; (12.6) where: Z XI D†jZj XI , e X i Dh e X i 1 : : : e X i N i T y D 1 : : : N T In (12.6) the vector of Lagrange multiplier, *, is the minimizer of the double optimization problem proposed in (12.4b) Dargmin 2RN lnZ.XI / (12.7) Which leads to the following system of nonlinear equations f . / Dr lnZ. / D N X jD1 !j .X/ e X i D0; (12.8) where r represents the gradient with respect to . Once convergence * is found, the weighting functions are calculated in (12.6).
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